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In a series system, the system fails if any of the components fails. When the system functions, there may exist correlation among components because they are connected within the same system. In this paper, we consider the reliability analysis of multiple Type-I censored life tests of series systems composed of two components with bivariate log-normal lifetime distributions. The major interest is the inference on the mean lifetimes, and the reliability functions of the system and its components. Given observations of the minimum lifetime of the components of each failed system, location of the MLEs highly relies on the initial values in executing the computation numerically. Alternatively, we apply the Bayesian approach after a re-parametrization of the parameters of interest. A simulation study is conducted which shows that the Bayesian approach provides considerably accurate inference. The proposed approach is successfully applied to a real data set.